Some properties of q-Gaussian distributions

Abstract

In this research article, we introduced the notion of q-probabilty distributions in quantum calculus. We characterized the concept of q-density by connecting it to a probability measure and investigated some of their outstanding properties. In this case, the Transfer theorem was extended in order to compute afterwards the q-moments, q-entropy, q-moment generating function, and q-quantiles. We are also interested in finding the centered q-Gaussian distribution ${\mathcal{N}}_q(0,{\sigma }^2)$ with variance ${\sigma }^2$. We also proved that this q-distribution belongs to a class of classical discrete distributions. The centered q-Gaussian law ${\mathcal{N}}_q(0,{\sigma }^2)$ is also naturally related to the q-Gaussian distribution ${\mathcal{N}}_q(\mu ,{\sigma }^2)$ with mean μ and standard deviation σ. We corroborated that the q-moments of these q-distributions are q-analogs of the moments of classical distributions. Numerical studies demonstrated that ${\mathcal{N}}_q(0,{\sigma }^2)$ interpolates between the classical Uniform and Gaussian distributions when q goes to 0 and 1, respectively. Subsequently, simulation studies for various q parameter values and samples sizes of the Gaussian q-distributions were conducted to demonstrate the effectiveness of the proposed model. Eventually, we provided some pertinent closing remarks and offered new perspectives for future works.

KEYWORDS:

• q-Calculus
• q-Gaussian distribution
• q-transfer theorem
• q-entropy
• q-quantile

Acknowledgments

We would like to thank the Editor-in-Chief of Communications in Statistics–Theory and Methods, Prof. Narayanaswamy Balakrishnan, and the three anonymous reviewers for their helpful comments, which helped us to focus on improving the original version of the article.

Disclosure statement

No potential conflict of interest was reported by the authors.

Table 1 The q-Quantiles of different order of ${\mathcal{N}}_q(0,0.5)$ for different values of q.

Additional information

Funding

Authors acknowledge financial support from the Hubert Curien “PHC-Utique” program (CMCU number: 20G1503–Campus France number: 44172SL), implemented by Campus France, as well as the project CPER “E-Data” funded by the Nouvelle-Aquitaine Region. The Region and the European Union are also supporting this project within the framework of the “Programme Opérationnel FEDER/FSE 2014-2020”.